Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500120 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2709 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2)(3,4)(21,22,25,30,26,29,38,43)(23,31,33,40,34,44,37,27)(24,28,36,42,35,41,48,51)(32,45,47,50,46,52,49,39);(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,23,26,34)(22,27,29,40)(24,32,35,46)(25,37,38,33)(28,39,41,50)(30,44,43,31)(36,49,48,47)(42,52,51,45);(2,4)(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,24)(22,28)(23,32)(25,36)(26,35)(27,39)(29,41)(30,42)(31,45)(33,47)(34,46)(37,49)(38,48)(40,50)(43,51)(44,52);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(22,29)(27,40)(28,41)(30,43)(31,44)(39,50)(42,51)(45,52);(6,12)(10,13)(11,20)(18,19)(24,35)(28,41)(32,46)(36,48)(39,50)(42,51)(45,52)(47,49)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=h,d^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6436603);